1550251515-Classical_Complex_Analysis__Gonzalez_

66 Chapter 1

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  10. Let ds^2 = dx^2 + dy^2 and let d0"^2 = da^2 + d/3^2 + d1^2 be the corresponding
      differential of arc on the sphere. Show that
      ds
      dO" = 1 + lzl2
  
  
  
  

1.18 THE GENERAL BINARY COMPLEX NUMBER

We have seen (Section 1.2) that the definition of multiplication of ordinary

complex numbers is equivalent to the property i^2 =:= -1, provided that the

validity of the usual multiplication laws be granted. We may ask what kind

of algebraic system will result by assuming, more generally, that

i^2 = Oi + i/3 (1.18-1)

where a and /3 are given real numbers, and keeping the definitions of equal-

ity and addition of two complex numbers as in Definition 1.1. Under

assumption (1.18-1) the imaginary unit i = (0, 1) is then a root of the

quadratic equation

t^2 - j3t - Oi = 0 (1.18-2)

which reduces to t^2 + 1 = 0 in the usual case, i.e., for a = -1, /3 = 0. By

applying the multiplication laws (which we wish to uphold) and making

use of (1.18-1), we find that the product of any two complex numbers now

will be given by

(a + bi)( c + di) = ( ac + bda) + (ad+ be+ bd/3)i

As shown by P. Capelli [2a], the properties· of this general complex

system depend essentially on the value of the discriminant

.6. = 132 + 4a

of the equation (1.18-2). If .6. < 0, the system is called elliptic and it

is isomorphic to the ordinary complex system. In fact, an elliptic system

always contains an element j = (/3 -2i)/F/I such that

J.^2 - /3^2 - 4/3i + 4i^2 - /3^2 + 4a - -1

• -.6. - -.6. -

From the definition of j we get i = (/3 - F/Ij)/2, so

b(/3-F/S)j

a + bi = a + -'-----'----

2

= (a +^1 / 2 b/3) -^1 / 2 bV-E,j = A + B j (1.18-3)